Weighted Birkhoff averages

 Given a dynamical system $(X,T)$ and a potential $\phi:X\to\R$, the Birkhoff average of $\phi$ at a point $x\in X$ is defined as

\[
\overline{\phi}(x) = \lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} \phi(T^i(x)),
\]
wherever the limit exists. This object is interesting from many points of view, the point of view that is especially interesting to me is the multifractal description of the limit sets of $\overline{\phi}$.

Lately people started to investigate versions of Birkhoff average which involve adding some weights. There are two versions of such weighted Birkhoff averages, both of them appear in particular in the investigation of the Sarnak conjecture in the ergodic theory.

Version I: let $(w_i)$ be a prescribed sequence of bounded reals, and let

\[
\overline{\phi}_1(x) = \lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} w_i \phi(T^i(x)).
\]

Version II: let $(w_i)$ be a decreasing non-summable sequence of positive reals, and let

\[
\overline{\phi}(x) = \lim_{n\to\infty} \frac 1 {\sum_{j=0}^{n-1} w_j} \sum_{i=0}^{n-1} w_i \phi(T^i(x)).
\]

In the talk I will present some results (obtained with Balazs Barany and Ruxi Shi) on those kinds of weighted Birkhoff averages and their multifractal properties.